In their paper, the authors W. Dembski, W. Ewert, and R. Marks (DEM) talk about something they call the natural probability:

Processes that exhibit stochastic behavior arise from what may be called a natural probability. The natural probability characterizes the ordinary stochastic behavior of the process in question. Often the natural probability is the uniform probability. Thus, for a perfect cube with distinguishable sides composed of a rigid homogenous material (i.e., an ordinary die), the probability of any one of its six sides landing on a given toss is 1/6. Yet, for a loaded die, those probabilities will be skewed, with one side consuming the lion’s share of probability. For the loaded die, the natural probability is not uniform.

This natural probability on the search space translates through their idea of lifting to the space of measures $\mathbf{M}(\Omega)$:

As the natural probability on $\Omega$, $\mu$ is not confined simply to $\Omega$ lifts to $\mathbf{M}(\Omega)$, so that its lifting, namely $\overline{\mu}$, becomes the natural probability on $\mathbf{M}(\Omega)$ (this parallels how the uniform probability $\mathbf{U}$, when it is the natural probability on $\Omega$, lifts to the uniform probability $\overline{\mathbf{U}}$ on $\mathbf{M}(\Omega)$, which then becomes the natural probability for this higher-order search space).

As usual, I look at an easy example: a loaded coin which always shows head. So $\Omega=\{H,T\}$ and $\mu=\delta_H$ is the natural measure on $\Omega$. What happens on $\mathbf{M}(\Omega)= \{h\cdot\delta_H + t\cdot\delta_T|0 \le h,t \le 1; h+t=1 \}$? Luckily,
$$(\mathbf{M}(\{H,T\}),\mathbf{U}) \cong ([0,1],\lambda).$$
Let's jump the hoops: